Computer Science > Computational Complexity

Title:
On the sum of the L1 influences of bounded functions

Abstract: Let $f\colon \{-1,1\}^n \to [-1,1]$ have degree $d$ as a multilinear
polynomial. It is well-known that the total influence of $f$ is at most $d$.
Aaronson and Ambainis asked whether the total $L_1$ influence of $f$ can also
be bounded as a function of $d$. Ba\v{c}kurs and Bavarian answered this
question in the affirmative, providing a bound of $O(d^3)$ for general
functions and $O(d^2)$ for homogeneous functions. We improve on their results
by providing a bound of $d^2$ for general functions and $O(d\log d)$ for
homogeneous functions. In addition, we prove a bound of $d/(2 \pi)+o(d)$ for
monotone functions, and provide a matching example.

Comments:

16 pages; accepted for publication in the Israel Journal of Mathematics